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In the entry on hypercovers, the codomain is assumed to be representable, and the notion thus depends on the site used (?). In some source the setting is a locally connected topos and then hypercovers are defined in a way that seems more general and the n-th level mapping are (simply) epic. Are they equivalent? I ask because I want to write an entry summarising the Artin-Mazur etale homotopy type construction in an attempt to clarify (for myself!) some of the points raised in earlier discussions. (I feel I should know all this off by heart but find it still unclear.)
I think that although the entry is clearer than it was originally, there are intuitions abut why hypercoverings rather than coverings that are somehow missing (and are not clear to me), e.g. that hypercoverings are fibrant objects, the condition generalises the Kan condition, etc. If I can sort this out in my own mind I will adjust the entry accordinly, but this does need me to be clearer than I am at present!
Well, a notion which requires the codomain to be representable and the covers to be coproducts of representables certainly isn’t going to be exactly the same as one which doesn’t. I would, though, expect that any hypercover defined “internally” in a topos of sheaves could be refined by one which consists of representables. In particular, I’m pretty sure that a map of sheaves is epic in the category of sheaves iff it is a “locally epimorphism” of presheaves.
The difference is probably analogous to the difference between (1) a covering family in a site, which is equivalent to giving a local epimorphism in the presheaf category whose codomain is representable and whose domain is a coproduct of representables (which is, I think, the same as a height-1 hypercover in the definition given on the page) and (2) an arbitrary local epimorphism in a presheaf category.
BTW, I’m not hugely fond of the definition that involves coproducts of representables in a presheaf category. I generally feel like it makes notions clearer to formulate them purely in terms of covering families in the site. In those terms, it seems like a hypercover would be a particular kind of a diagram in the site whose shape is the category of simplices of some simplicial set (probably a contractible Kan complex). Is a definition along those lines written down anywhere?
One can define hypercovers of a constant simplicial object A in any finitely complete site with a singleton pretopology: one just takes an internal Kan fibration where the surjections in the definition of a Kan fibration are replaced by covers. I believe a local epimorphism in a presheaf category can be refined by a cover which is a coproduct of representables, so there shouldn’t be any difference in the end result if we take our covers to be local section admitting maps associated to either of these two. I’m pretty sure a map of sheaves is epic if it is a local epimorphism (does this follow from the fact sheafification is a left adjoint and hence preserves epimorphisms?)
The full story is given in that reference by Jardine that the entry points to. I’ll spell it out now.
Okay, I generalized the definition in the entry to the notion of hypercovers over arbitrary (even simplicial) objects as local acyclic Kan fibrations.
Also restructured slightly and added an explicit statement of Verdier’s hypercovering theorem. (This follows as a corollary of the model category results by Dugger-Hollander-Isakson, but is still of interest in its own right.)
@Urs Thanks. That clears up my difficulty no end.
That clears up my difficulty no end.
Okay, good. One can rewrite what I wrote more intrinsically by equivalently just talking about epimorphisms in the sheaf topos. Probably that’s eventually a better way to say it. But I have to look into something else right now.
In fact I like the way you put it. It says the machinery of presheaf homotopy reflects down exactly in this way and so interprets hypercoverings neatly.
The definition of bounded hypercover seems to be missing something.
The definition of bounded hypercover seems to be missing something.
The formula for the morphism was still the old one for the assumption that the domain is representable. I fixed that now. Is this what you mean? Or do you think something else is missing?
I think there was also a variable missing from the codomain, which confused me. Now it looks good, thanks.
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